Principles of Managerial Finance

The future value with continuous compounding therefore equals $117.35. As

expected, the continuously compounded value is larger than the future value of

interest compounded semiannually ($116.99) or quarterly ($117.16). Continu-

ous compounding offers the largest amount that would result from compounding

interest more frequently than annually.

Nominal and Effective Annual Rates of Interest

Both businesses and investors need to make objective comparisons of loan costs

or investment returns over different compounding periods. In order to put inter-

est rates on a common basis, to allow comparison, we distinguish between nomi-

nal and effective annual rates. The nominal,or stated, annual rateis the contrac-

tual annual rate of interest charged by a lender or promised by a borrower. The

effective, ortrue, annual rate (EAR)is the annual rate of interest actually paid or

earned. The effective annual rate reflects the impact of compounding frequency,

whereas the nominal annual rate does not.

Using the notation introduced earlier, we can calculate the effective annual

rate, EAR, by substituting values for the nominal annual rate, i, and the com-

pounding frequency, m,into Equation 4.23:

EAR 1  

m

 1 (4.23)

We can apply this equation using data from preceding examples.

i

m

effective (true) annual rate (EAR)
The annual rate of interest
actually paid or earned.

180 PART 2 Important Financial Concepts

nominal (stated) annual rate
Contractual annual rate of
interest charged by a lender or
promised by a borrower.

117.35

0.16 2 nd

ex



(^100) 
1.1735
Solution
Input Function
EXAMPLE To find the value at the end of 2 years (n2) of Fred Moreno’s $100 deposit
(PV$100) in an account paying 8% annual interest (i0.08) compounded
continuously, we can substitute into Equation 4.21:
FV 2 (continuous compounding)$100e0.08^2
$1002.71830.16
$1001.1735$117.35
Calculator Use To find this value using the calculator, you need first to find the
value ofe0.16by punching in 0.16 and then pressing2ndand thenexto get 1.1735.
Next multiply this value by $100 to get the future value of $117.35 as shown at the
left. (Note:On some calculators, you may not have to press2ndbefore pressingex.)
Spreadsheet Use The future value of the single amount with continuous com-
pounding also can be calculated as shown on the following Excel spreadsheet.